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Question
Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.
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Solution
Let the given statement be P(n), i.e.,
P(n):41n – 14nis a multiple of 27.
It can be observed that P(n) is true for n = 1 since `41^1 - 14^1 = 27`, which is a multiple of 27.
Let P(k) be true for some positive integer k, i.e.,
41k – 14kis a multiple of 27
∴41k – 14k = 27m, where m ∈ N … (1)
We shall now prove that P(k + 1) is true whenever P(k) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.
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